--- title: "The Sine Rule" description: "Use a/sin A = b/sin B in triangles with no right angle to find a missing side or angle, and check whether an obtuse second answer also fits." canonical: https://lightmysky.com/learn/mathematics/the-sine-rule-mt_cTt_-9KZho source: https://lightmysky.com/learn/mathematics/the-sine-rule-mt_cTt_-9KZho.md retrieved: 2026-11-12 --- > **Agent view.** This is the Markdown twin of the page, for tools and assistants. > When to use this site, and the call that answers each job: https://lightmysky.com/agent-instructions.md > API description (OpenAPI 3.1): https://lightmysky.com/openapi.json · Authentication: https://lightmysky.com/auth.md > Pricing: https://lightmysky.com/pricing.md · Catalog: https://lightmysky.com/llms.txt · Full catalog: https://lightmysky.com/llms-full.txt > Every machine-readable file on this domain: https://lightmysky.com/.well-known/ai-catalog.json > Ask for Markdown with `Accept: text/markdown`, a `.md` address, or `?mode=agent`. # The Sine Rule Use a/sin A = b/sin B in triangles with no right angle to find a missing side or angle, and check whether an obtuse second answer also fits. Subject: Mathematics · Area: Geometry · Ages 15 to 16 Page: https://lightmysky.com/learn/mathematics/the-sine-rule-mt_cTt_-9KZho ## Ready when they can - Pair each side with the angle opposite it before substituting - Find a missing angle and test whether the obtuse partner also works - Choose the sine rule when a side and its opposite angle are both known ## Lesson: Pair the side with its angle The sine rule works for any nondegenerate triangle, right angled or not: a over sin A equals b over sin B equals c over sin C. Before you substitute anything, pair each side with the angle opposite it. Reach for this rule when a side and its opposite angle are both known, plus another known side or angle. Switch to the cosine rule when two sides and the included angle are known, or when all three sides are known. *(drawing: Side a pairs with angle A, side b pairs with angle B, side c pairs with angle C.)* Finding an angle hides one trap. Rearrange the sine rule to sin B equals b times sin A over a, then take the inverse sine. Sine is shared by supplementary angles, so your calculator hands you the principal value, which is acute or exactly 90 degrees. Always test 180 minus the calculator value too: 30 and 150 share a sine, and 90 is its own supplement. Check both candidates against every given fact: the third angle must stay positive, and any stated angle restrictions must hold. Never pick an answer just because a sketch looks obtuse. **Example.** Try it: angle A is 50 degrees with opposite side 7 cm, and angle B is 80 degrees. Pair a with A and b with B, then write b over sin 80 equals 7 over sin 50. Since sin 80 is about 0.985 and sin 50 is about 0.766, b equals 7 times 0.985 over 0.766, which is about 9.0 cm. **Tip.** When no full side-angle pair exists, stop and check what else is given. A missing angle may come from the 180 degree sum, or the data may be insufficient. And always label the triangle before touching any formula: pairing a side with its neighbour rather than its opposite is a common error to check for. **Recap.** Pair sides with opposite angles, test the obtuse partner against every given fact, and check the data is sufficient before choosing a rule. ## Practice 14 questions on this page, each with its working shown. ## Needs first - [Exact Values of Sine, Cosine and Tangent](https://lightmysky.com/learn/mathematics/exact-values-of-sine-cosine-and-tangent-mt_SNJEbXYOrR) ## Opens up - [Combining a sine and a cosine into a single wave](https://lightmysky.com/learn/mathematics/combining-a-sine-and-a-cosine-into-a-single-wave-mt_IwQkNDRWyq) - [Bearing Problems with the Sine and Cosine Rules](https://lightmysky.com/learn/mathematics/bearing-problems-with-the-sine-and-cosine-rules-mt_YQFdU4aiFP) - [The Cosine Rule](https://lightmysky.com/learn/mathematics/the-cosine-rule-mt_yy73gK5CEP)